Full text: Barrow, Isaac: Lectiones Opticæ & Geometricæ

lata, pèrque motum iſtum in curva deſcribenda conſpirans, percurrit
rectam PM. Cùm igitur ſint TP, PM ex conſtructione pares,
adeóque velocitates motuum, quibus ſimul peraguntur, æquales; etiam motus deſcenſivus in P, vel M æquabitur motui tranſverſo, cur-
vam deſcribenti, hoc eſt motûs ab S ad A velocitas in A eidemæquatur. Ergo punctum S eſt id ipſum, quod inveniri debuit, & abſolutum eſt
propoſitum. | Exemplo ſit _parabola_, quæ facta concipitur ex motu
uniformi horizontali, & deſcenſivo pariter accelerato; tum punctum
P ità facilè per _Analyſin_ inveſtigatur. Sit recta R _datæ parabo [?] læ_
_rectuns [?] latus._ Eſt igitur ex _parabolæ_ natura, R x AP. = PMq
= TPq (exhypotheſi modi noſtri generalis.) Item, ex parabolæ
nota proprietate eſt TPq = 4 APq. Ergo eſt R x AP = 4 APq. Adeóque R = 4AP; vel {1/4} R = AP = SA. Nimirum ita _Gali-_
_læus_ determinavit. In hoc autem caſu puncta T, S coincidunt. Quòd
ſi rurſus gravia juxta _triplicatam temporum rationem_ velocitate creſcen-
do deſcendant, adeóque motus ipſorum talis cum uniformi tranſverſo
compoſitus _parabolam cubicam_ deſcribat, & ſit R iſtius curvæ _para-_
_meter_, erit eo in caſù SA = √ {R q/27} nam ex hujuſce curvæ proprie-
tate eſt R q AP = PM cub. Et ex hujus regulæ generalis præſcripto
eſt PM = TP, adeóque PM cub. = TP cub. Denique quoniam
in hujuſmodi _parabola_ tangentis intercepta ſemper triſecatur à vertice
(nimirum ut ſit AP = {1/3} TP) eſt TP cub. = 27 AP cub. Erit
igitur R q AP = 27 AP cub. Adeóque R q = 27 APq; vel
{Rq/27} = APq = SAq. In reliquis ſimili ratione procedentes
aſſequemur propoſitum. Poſſent opinor & hinc nedum pleræque
_Galilæipoſitiones_ huic affines, & hanc attingentes materiam utcun-
que deduci, ſed & generaliores reddi, vel ad alia curvas omnigenas
extendi. Verùm parco pluribus, hoc _ſpecimine_ (quoad iſta) con-
tentus; huc non niſi per tranſcurſum adducto. Ad alia pergo præ-
dictis cohærentia.

31.1.

Fig. 22.
Note:

XVI. Si ad rectam lineam applicetur _planæ ſuperficies_, cujus
ſingulæ quæque partes applicatis ad iſtam rectam parallelis inter-
ceptæ proportionales ſint rectis ad rectam AY ſimpliciter diviſam
applicatis (ad AZ nempe parallelis.) Hujuſce ſuperficiei ad paral-
lelogrammum æquealtum, ſuper eadem baſe conſtitutum, proportio
proportionem indicabit ipſarum AP; TP, à puncto P vertici, tan-
gentique interjectarum.

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